A. Verhoeven

Model order reduction for nonlinear IC models

Model order reduction is a mathematical technique to transform nonlinear dynamical models into smaller ones, that are easier to analyze. In this paper we demonstrate how model order reduction can be applied to nonlinear electronic circuits. First we give an introduction to this important topic. For linear time-invariant systems there exist already some well-known techniques, like Truncated Balanced Realization. Afterwards we deal with some typical problems for model order reduction of electronic circuits. Because electronic circuits are highly nonlinear, it is impossible to use the methods for linear systems directly. Three reduction methods, which are suitable for nonlinear differential algebraic equation systems are summarized, the Trajectory piecewise Linear approach, Empirical Balanced Truncation, and the Proper Orthogonal Decomposition. The last two methods have the Galerkin projection in common. Because Galerkin projection does not decrease the evaluation costs of a reduced model, some interpolation techniques are discussed (Missing Point Estimation, and Adapted POD). Finally we show an application of model order reduction to a nonlinear academic model of a diode chain.

Adjoint transient sensitivity analysis in circuit simulation

Sensitivity analysis is an important tool that can be used to assess and improve the design and accuracy of a model describing an electronic circuit. Given a model description in the form of a set of differential-algebraic equations it is possible to observe how a circuits output reacts to varying input parameters, which are introduced at the requirements stage of design. In this paper we consider the adjoint method more closely. This method is efficient when the number of parameters is large.We extend the transient sensitivity work of Petzold et al., in particular we take into account the parameter dependency of the dynamic term.We also compare the complexity of the direct and adjoint sensitivity and derive some error estimates. Finally we sketch out how Model Order Reduction techniques could be used to improve the efficiency of adjoint sensitivity analysis.

Model Order Reduction for Nonlinear Differential Algebraic Equations in Circuit Simulation

In this paper we demonstrate model order reduction of a nonlinear academic model of a diode chain. Two reduction methods, which are suitable for nonlinear differential algebraic equation systems are used, the trajectory piecewise linear approach and the proper orthogonal decomposition with missing point estimation.

Stability analysis of the BDF Slowest-first multirate methods

This paper deals with the stability analysis of BDF Slowest-first multirate time-integration methods applied to the transient analysis of circuit models. From an asymptotic analysis it appears that these methods are indeed stable if the subsystems are stable and weakly coupled.

Automatic partitioning for multirate methods

The (nonlinear) transient analysis of electrical circuit models plays an important role in circuit design. Multirate time integration can be able to achieve the same accuracy for much lower costs. An essential assumption is the existence of a good partition of the circuit in a slow and fast part. This paper describes how this can be done automatically.

Proper orthogonal decomposition model order reduction of nonlinear IC models

We demonstrate Model Order Reduction for a nonlinear system of differential-algebraic equations of a diode chain by Proper Orthogonal Decomposition with Adapted Missing Point Estimation. The collected time snapshots also allow for an efficient impression of the sensitivity of objective functions.

Aspects of Multirate Time Integration Methods in Circuit Simulation Problems

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MODEL ORDER REDUCTION: AN ADVANCED, EFFICIENT AND AUTOMATED COMPUTATIONAL TOOL FOR MICROSYSTEMS

The goal of mathematical model order reduction (MOR) is to replace the non-automatic compact modeling, which is the state of the art in simulation flow of microelectronic and micro-electromechanical systems (MEMS). MOR offers a possibility of automatically creating small but very accurate models which can be used within system level simulation. The main challenges in integrating model order reduction as a standard tool in the current simulation flow are to be able to reduce non-linear ordinary differential equation systems (ODEs) and differential algebraic equation systems (DAEs), which arise from either spatial discretization of partial differential equations (PDEs) or from electronic circuit equations. We present a methodology for applying MOR to linear and non-linear ODEs and DAEs and numerical results for several MEMS and microelectronic devices.

Model order reduction for nonlinear IC models with POD

Due to refined modelling of semiconductor devices and increasing packing densities, reduced order modelling of large nonlinear systems is of great importance in the design of integrated circuits (ICs). Despite the linear case, methodologies for nonlinear problems are only beginning to develop. The most practical approaches rely either on linearisation, making techniques from linear model order reduction applicable, or on proper orthogonal decomposition (POD), preserving the nonlinear characteristic. In this paper we focus on POD. We demonstrate the missing point estimation and propose a new adaption of POD to reduce both dimension of the problem under consideration and cost for evaluating the full nonlinear system.

Digital Linear Control Theory Applied To Automatic Stepsize Control In Electrical Circuit Simulation

Adaptive stepsize control is used to control the local errors of the numerical solution. For optimization purposes smoother stepsize controllers are wanted, such that the errors and stepsizes also behave smoothly. We consider approaches from digital linear control theory applied to multistep BDF-methods.